Invariance and scaling properties in the distributions of contributing area and energy in drainage basins

The cumulative probability distributions for stream order, stream length, contributing area, and energy dissipation per unit length of channel are derived, for an ordered drainage system, from Horton's laws of network composition. It is shown how these distributions can be related to the fractal nature of single rivers and river networks. Finally, it is shown that the structure proposed here for these probability distributions is able to fit the observed frequency distributions, and their deviations from straight lines in a log‐log plot.